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An economic analysis of apathetic behavior: Theory and experiment
The Journal of Socio-Economics 37 (2008) 90–107
An economic analysis of apathetic behavior: Theory and experiment夽 Aiko Shibata a,∗ , Toru Mori b , Makoto Okamura c , Noriko Soyama d a
d
Japanese Fair Trade Commission, Kasumigaseki, Tokyo, Japan b Faculty of Economics, Nagoya City University, Japan c Economics Department, Hiroshima University, Japan Center of Research and Development of Liberal Arts Education, Tenri University, Japan Accepted 1 December 2006
Abstract The apathy of bystanders often prevails when instances of bullying, hidden crime and extortion occur in communities such as schools, business work areas, underclass ghettos, prisons and the military. The present study models apathetic behavior as a non-cooperative game and attempts to verify this theory through experiments. Furthermore, our research suggests that the apathy of bystanders generally decrease as the number of citizens in a community decrease. In our experimental cases, if the number of members in a group decreases from 40 members to 20 members, the concerned and helpful behavior of bystanders increases by 21%. © 2007 Elsevier Inc. All rights reserved. JEL classification: C92; H41 Keywords: School bullying; Workplace bullying; Bystanders’ behavior; Tattling behavior; Non-cooperative games; Dynamic adjustment process; Public goods; Experimental economics
1. Introduction Bullying can contribute to an environment of fear and intimidation in school and other places, and has become serious matters in the world (Olweus et al., 1999). For example, according to National Conference of State Legislatures in the U.S.A., 17 states enacted school safety laws
夽 ∗
Authors’ names are listed by seniority. Corresponding author. E-mail address: [email protected] (A. Shibata).
1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2006.12.026
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concerning bullying and student harassment between 1994 and 2004. The OECD jointly with Norwegian Government and University held an International Policy Conference on School Bullying and Violence in 2004. According to a 1996-survey of 2900 middle and high school students by the Seoul Family Court in South Korea, about 90% of the students was physically threatened by other students (Ort, 1999). However, bystanders may not be willing to tattle the illegal activities to the school authorities. Bullying is also observed at workplace. The cost of violence and bullying at work is very large as well. For example, the UK nation-wide survey of workplace bullying reported that the bullied group was found to have higher absenteeism due to bullying and, the currently bullied group was found to have on average 7 days more off work in a year than those who were neither bullied nor had witnessed bullying taking place. Those who are bullied also decide to leave their organization as a result of their experience. This example of bullying in the UK could be appropriate for other countries as well (Martino, 2002; Hoel et al., 2001). There are similar situations as bullying. In crime-ridden areas, people are extremely reluctant to testify against those whom they have observed robbing and killing because they expect that the criminals will not go to jail and will punish any witnesses that testify. Another situation may be political revolution. If you revolt and not enough others join you, you will be punished. If the number of people who revolt reaches a threshold, the revolt will succeed and revolutionaries will be rewarded rather than punished. The purpose of this paper is to address this question of why the apathy of bystanders predominates despites their compassionate feelings1 . To this end, we formulate the phenomenon of bullying in a school as a non-cooperative n person game. This game is a kind of binary choice public goods game with threshold. We will also examine the relevance of our model by experiment. The rest of paper is organized as follows: Section 2 formulates the model. Section 3 derives three type Nash equilibria and examines the stability of the equilibrium by introducing a dynamic adjustment process. We consider policy implications in Section 4. Experiment study is reported in Section 5. Final section gives concluding remarks. 2. The Model We consider bullying in school. A schoolyard has a bully who does bad stuff that is observed by the n other children but not by the teacher. The teacher will punish the bully if at least t of the children tattle on him. If she does not tattle and fewer than t other people tattle, she pays a cost b of being annoyed by continued harassment from the bully in a schoolyard. If she does tattle and fewer than t − 1 other people tattle, she should pay not only the cost b but an extra cost c (perhaps the bully gives her extra harassment as punishment for her tattling). If at least t people in the class tattle, then the bully is stopped. In this case nobody in the class has to pay the cost b, nor do those who tattled have to pay any cost for tattling (perhaps the bully is sent to reform school and cannot punish the squealers). The unusual, but perhaps often realistic, feature
1 Bystanders may not be willing to report the illegal activities to the school authorities. The reason for this may be explained as follows. The students’ recognition of the illegal instances which occur in their classroom is higher by several percentage points than the teachers’ recognition of the same instance (Rigby and Slee, 1991). But this number reverses itself when the actual reports of instances are counted. According to a survey of 20,000 students carried out by the Ministry of Education in Japan (1994/1996), the rate of discovery of instances of bullying by teachers is two to three times higher than the rate of discovery based on reports of students.
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Table 1 A representative player’s payoffs when she selects two strategies, the not tattling strategy S and the tattling strategy R Numbers of other tattlers
0
S (not-tattle) R (tattle)
w−b w−b−c
···
t−2
t−1
t
w−b w−b−c
w−b w
w w
···
n−1 w w
The top row shows the number of other people who tattle and w is the initial utility level when there is no disorderly activity in a community.
of the payoff structure is that tattling is costly only if there are not enough tattlers to guarantee enforcement.2 We model the above story as a n player non-cooperative game. The players are n students. Each player (student) has two strategies: tattling (R) and not tattling (S). There is a threshold value t (0 ≤ t ≤ n) in this game. The payoff of a player is illustrated in Table 1. We can interpret this game as follows: in this game, a status quo is a situation with continuous illegal activity (disorder) generating negative externalities. To eliminate the negative externalities by tattling can be interpreted to provide a public good. The tattling cost c corresponds to a provision cost of the public good. The strategy R(S) shows the concerned and helpful (apathetic) behavior. Then, this game belongs to n person, binary-choice, public good supply game. Sandler (1992) classifies this public good game into six categories, Prisoner’s Dilemma, Chicken, Coordination, Assurance 1 and Assurance 2. The game we define is in the category of Assurance 2, which describes the provision of a discrete public good without refunding costs of players. Minimum efforts must be provided to receive any benefit resulting from the public good and the associated costs are solely assigned to each provider (selecting the R strategy). On the other hand, costs are shared among all players in Assurance 1. The literature includes Hirshleifer (1983), De Jasay (1989), Gardner et al. (1990), Palfrey and Rosenthal (1984), Runge (1984) and Schelling (1973). The game we construct, however, differs from the Assurance game in two points. First, in our game, each player does not need to incur the costs once the negative externalities disappears in the class, while, in the Assurance 2, the cost has to be paid by a player at any situation. This makes a mirror image of the game with refund that each player is not required to pay the cost unless the public good is provided. Second, the benefit each player receives does not vary across the number of contributors, whilst this benefit rises with that of contributors in the Assurance 2. The structure of equilibrium is, however, essentially same as that of Assurance 2. 3. The Nash equilibrium and its dynamic stability First, we derive the Nash equilibrium of the game. In our game, every player is assumed to have an identical utility (payoff) function. The utility of a player depends on not only her strategy but also on the strategies chosen by the other players. The crucial determinant of a player’s utility is the number of other players who tattle the disorderly activity. The combination of a representative player’s and others’ strategies are listed in the pre-stated Table 1. Let us consider a typical player and assume that she thinks that the probability of the other players’ taking the R strategy is q. Then, 2 Here, we assume that an extra cost c of tattling is the costs of possible retaliation. Because time cost for tattling is very small as tattling could be performed at any time when a tattler chooses at his convenience and a traveling cost to a place for tattling is also very small as tattling happens at a place where one visits regularly, such as work places and school buildings.
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the expected utility of this player when she chooses the strategy R(S) is given by ER (q)[ES (q)]. ER (q) = (w − b − c)Z(t − 2) + w [1 − Z(t − 2)] = w − bZ(t − 2) − cZ(t − 2)
(1)
ES (q) = (w − b)Z(t − 1) + w [1 − Z(t − 1)] = w − bZ(t − 1)
(2)
where Z(k) is the sum of probabilities that the number of other players who take the R strategy is less than or equal to k. For example, t−2 n−2 Z(t − 2) = qk (1 − q)n−2−k (3) k k=0 Z(t − 1) =
t−1 k=0
n−1 k
qk (1 − q)n−1−k
By substituting these values into (1) and (2), we obtain as follows: t−2 n−2 qk (1 − q)n−2−k ER (q) = w − (b + c) k k=0 Es(q) = w − b
t−1 k=0
n−1 k
(4)
(5)
qk (1 − q)n−1−k
(6)
Since Z(t − 1) > Z(t − 2) from (3) and (4), this difference in the two probabilities reflects the difference in the threshold number of tattlers from the member’s point of view. If she tattles to the authority, the threshold number t − 1, is the minimum required number of tattling by others necessary to resolve the continuous illegal activity in the community. If she does not tattle to the authority, the threshold number remains t. In Eq. (5), the level of utility w is decreased both by the expected amount b of the negative externality and by the expected private cost c. In Eq. (6), the level of utility w is decreased only by the expected amount b of the negative externality. However, the corresponding probabilities in the two equations are different. At the mixed strategy equilibrium, both expected utilities should be equal, i.e., ER (q(t)) = ES (q(t)). By rewriting this relationship, we obtain the following: b Z(t − 2) = . c Z(t − 1) − Z(t − 2)
(7)
Then, we have the following proposition. Proposition 1. There are exactly two pure strategy Nash equilibria: all players tattle E(1) and none tattle E(0). And one symmetric mixed strategy equilibrium E(q* ) exists. Proof. Appendix A is referred.
Let us write the expected utility at equilibria E(0), E(1) and E(q* ) as U(0), U(1) and U(q* ), respectively. We then have the following relation between them: U(0) < U(q∗ ) < U(1)
(8)
Eq. (8) shows that the equilibrium E(1) where all players tattle dominates the other equilibria in the Pareto sense.
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Fig. 1. The structure of the players’ expected payoffs.
The proof of this proposition may become more transparent by showing Fig. 1 and then justifying its qualitative character. Fig. 1 shows the graphic expression of ER (q) and ES (q) which represent the player’s expected utilities from tattling and not tattling, respectively, and the graphic expression of them was obtained by applying specific numbers to variables in Eqs. (5) and (6) for ER (q) [ES (q)].3 The vertical axis measures the expected utilities and q is a probability of the other players’ tattling. For example, if the probability of tattling by the other players is greater than q* , then, ER (q) > ES (q), thus, the player is better off to tattle. On the other hand if the probability is less than q* , the reverse is true. ER (q) < ES (q) and the player is better off not to tattle. E(1) and E(0) are two pure strategy equilibria, all tattle and none tattle, respectively. The dynamic process converges to E(1)[E(0)] when the initial value of q, q0 , is larger (less) than q* . Thus, we have two stable equilibria, E(0) and E(1), and one unstable mixed strategy equilibrium, E(q* ). Let us call q* the critical probability as it is the dividing probability between converting all tattlers and none tattlers. By noting the following four facts, the qualitative character of Fig. 1 is justified: 1. 2. 3. 4.
ER (0) < ES (0). ER (1) = ES (1). There is exactly one q* ∈ (0,1) such that ER (q* ) = ES (q* ). The functions ER and ES are continuous.
From this, it follows trivially that ER (q) < ES (q) for all q < q* and ER (q) > ES (q) for all q ∈ [q* ,1), The only step of the above proof that is not immediately obvious is 3, and we have a proof 3 Fig. 1 is drawn from Eqs. (5) and (6) by assuming w = 150, b = 110, c = 10, n = 20, t = 15. These values are the values used for experiments.
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in Appendix A which could be read after the following section explaining our game in more detail. Next, we investigate which equilibrium players finally choose among the three Nash equilibria in the game. To tackle this problem, we formulate a dynamic adjustment process of the players’ strategies. Generally, the dynamic adjustment process indicating the change of q(m) can be formulated as Eq. (9) where t shows time and Φ(x) is a sign-preserving function, that is, Φ(x) 0 if and only if x 0. dq(m) = Φ ER (q(m)) − ES (q(m)) dt
(9)
We explain how the dynamic process (9) works. Suppose that two types of players, those who choose the pure strategy R, S, respectively, coexist in this game. We denote the number of players choosing the R strategy at a particular time t by r(m), 0 ≤ r(m) ≤ n. Each player can calculate her expected utility if she chooses pure strategy, R [S], for given r(m) as denoted by ER (r(m)) [ES (r(m))]. Since each player is assumed not to be perfectly-rational, she takes a time to change her strategy from S to R even though ER (r(m)) exceeds ES (r(m)) and vice versa. Consequently, only a fraction of players with less favorable strategies can switch their strategies to favorable one. Based on this idea, we can construct full dynamic adjustment process for three possible cases. (Case 1) :
ER (r(m)) > ES (r(m))
Players who choose the R strategy stick to this strategy because they enjoy the highest utility from the R strategy. A fraction of [n − r(m)] players who select the S strategy switch to the R strategy which gives them higher utility. Then the dynamic adjustment process in this case is given by dr(m) = β [n − r(m)] dt
(10)
where β is the fraction of players with changing strategy, 0 ≤ β ≤ 1. Since we can assume that q(m) = r(m)/n and n is fixed, the above process is rewritten as dq(m) = β 1 − q(m) dt
(11)
which belongs to (10). (Case 2) :
ER (r(m)) < ES (r(m))
Players with the S strategy continue to adopt their strategies. Since players with the R strategy are not satisfied, β fraction of them changes to the more advantageous strategy S. The adjustment process is defined by dr(m) = −βr(m), dt
(12)
that can be transformed to as follows: dq(m) = −βq(m) dt (Case 3) :
ER (r(m)) = ES (r(m))
(13)
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All players do not change their chosen strategies because they are indifferent between two strategies. Then, we have dr(m)/dm = dq(m)/dm = 0. Combining the adjustment processes (11) and (13) which belong to (9), we can construct full dynamic adjustment processes. Then, we establish the following: Proposition 2. Under the above dynamic adjustment process, (9), both the pure strategy Nash equilibria are globally stable, while the mixed strategy is globally unstable. Friedman and Fung (1996), for example, formulate the process which determines the distribution of two types of firms (American and Japanese) when each firm engages in Cournot competition in an autarkic or international market. Weibull (1995) provides an excellent survey and exposition of evolutionary game theory. 4. Policy implications In this section, we shall first discuss some theoretical issues and then present some policy implications for increasing bystanders’ concerned and helpful behavior. In our game-theoretical model presented in Section 3, we can show that the initial state of the game determines the final equilibrium. Since q0 is the probability at which players predict that the others choose the R strategy initially and is considered to be uniformly distributed, it is natural to consider q0 as an exogenous constraint. Thus, we assume that we cannot control the distribution of q0 . Then, the remaining procedure involves realizing the optimal equilibrium E(1) by decreasing the critical probability q* to as low a level as possible. The smaller the value of q* , the larger the probability that q0 is greater than q* , thus, the larger the possibility of the bystanders choosing the R strategy is at the initial round. This theoretical expectation is shown later to coincide with our experimental results. We examine what measures we need to lower q* . Since q* depends on four parameters {t, c, b, n}, it can be written as q* = q* (t, c, b, n). Thus, it is necessary to change the values of these four parameters in order to decrease q* . The relationship between q* and the other parameters can be specified by the following propositions. Proposition 3. 3-1 An increase in b or a decrease in c, hence an increase in b/c, reduces the value of q* . 3-2 A decrease in t decreases the value of q* . 3-3 Reducing n while keeping t/n unchanged, decreases the value of q* . Proof. See Appendix B.
Proposition 3-2 shows the effects of t on q* when n is given, while Proposition 3-3 illustrates the effects of n on q* when both n and t change, keeping t/n constant. Proposition 3-3 is based on the idea that it will be natural to think that the threshold number t changes proportionally to the number of members in a community, n. Then, Proposition 3-3 demonstrates that as the number of citizens in a community decreases, q* decreases, resulting in an increase in bystanders’ tattling activities. If the number of members in a group decreases from 40 members to 30 members and to 20 members, the critical probability, q* , decreases and the probability of resolving the continuing illegal activities in a community rises. For example, in our experimental cases, if the number of members in a group decreases from 40 members to 20 members, the concerned and helpful
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Fig. 2. Expected payoffs in the case where c is a negative value.
behavior of bystanders increases by 21% on average in our experimental cases.4 This decrease in the continuing illegal activities is not caused simply by the fact that it is easier for a supervisor or a teacher to supervise a smaller group. The decrease is rather caused by the expected utility maximizing behavior of the members in the community. Proposition 3-1 indicates that measures to increase b and to decrease c rises the chances of members tattling. Proposition 3-2 also means that by reducing the threshold number of tattling t, the concerned and helpful behavior increases. In the extreme cases in which c is either negative or a very large positive value, a Nash equilibrium with mixed strategy disappears. Suppose that c is negative. This situation is illustrated by the case in which a bystander is rewarded with sufficient compensation that exceeds her damages from retaliation when she chooses to tattle. For example, let us consider the situation in which the member, having tattled the illegal instances in the group, can receive a good evaluation for conduct because of her courage and honesty and in which this reward exceeds the costs of possible retaliation. In this case, the payoff obtained from tattling, w − b − c, is larger than w − b which is the payoff obtained by not tattling. That is, as described in Fig. 2 which is drawn from Eqs. (5) and (6) and variables remain the same as Fig. 1 except c which is changed to a negative value.5 The curve representing ER (q), the expected payoff in the case of tattling, runs above the curve representing ES (q), the expected 4 When the number of members in a community decreases from 40 to 20, an increase in cooperative behavior is calculated for each experimental case. For example, for Experiment E1: given t = 15, w = 120, b = 110, and c = 10, we obtain for n = 40, q* = 0.658 and for n = 20, q* = 0.585, thus, q* is reduced by 0.214 = [(1 − 0.585)/(1 − 0.658)] − 1. By calculating similarly the decreases of q* in eight experiments are 0.214 for Experiments E1, E3, E5, E8 and 0.195 for E4, E6, E7 and 0.241 for E2. The average and standard deviation are: average = 0.210, S.D. = 1.558E−02. 5 Fig. 2 is drawn from Eqs. (5) and (6) by assuming w = 150, b = 110, c = −10, n = 20, t = 15. Note that c is a negative value and represents a reward to a tattler.
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Fig. 3. Expected payoffs in the case where c is a large positive value.
payoff in the case of not tattling, for any q ∈ [0,1), and the curves coincide with each other only when q = 1. Thus, in the case that c < 0, our adjustment process leads to the only stable Nash equilibrium E(1), where all the players act as concerned and helpful. We consider, on the contrary, the case in which c takes a very large positive value. This implies the very high cost of retaliation as described in Fig. 3 which was drawn from Eqs. (5) and (6) and by changing c to a large positive value and other variables remain the same as Figs. 1 and 2.6 The curve representing ER (q), the expected payoff in the case of tattling, is always located below the curve representing ES (q), the expected payoff in the case of not tattling, for any q ∈ [0,1) and the curves cross each other only when q = 1. Thus, in this case, except for the unusual case in which all the bystanders decide to tattle the illegal incidences at the initial point in time, our adjustment process leads us, through a continuous reduction in the probability of tattling, to the only stable Nash equilibrium E(0), where all bystanders act apathetically. 5. Experimental analysis In this section, we examine the actual relevance of the adjustment process through experiment and show that the experimental results reasonably well explain the theoretical adjustment process.
6 Fig. 3 is drawn from Eqs. (5) and (6) by assuming w = 150, b = 110, c = 1000, n = 20, t = 15. Note that c represents a large cost to a tattler.
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5.1. Design of experiments We undertook 8 experiments (E1∼E8), each of which included 20 participants per group. Each participant played two consecutive sessions with ten rounds each using score tables in the form of Table 1 in Section 2. Participants in four experiments, E1, E2, E7 and E8, were from Kwansei Gakuin University and from Tezukayama University in the equal number, while in the other experiments, there were 18 Kwansei Gakuin, 16 Tezukayama and 6 Saitama University students. All participants were inexperienced undergraduate students. In each experiment, participants in the different universities were linked via a computer network. We used the system constructed by a client-server model on a TCP/IP network in E1 and E2, while we utilized the World Wide Web in the other cases.7 We did not inform the participants that the experiments were concerned with bullying problems nor any other issues. At the beginning of each session, subjects were told that they would be paid and at the end of each round, individual decisions and earnings were revealed to each player. However, the identities of subjects and where they sat were kept confidential to guarantee anonymity among subjects. The participants were given monetary rewards equal to their total scores after 20 rounds.8 Our experimental subjects were recruited from various majors at the three universities mentioned above. Subjects were randomly assigned to a booth with partitions in front and on both sides of the desk in the laboratory. It was impossible for them to make direct contact, i.e., by talking, making eye contact, with other subjects during the session. The taped instructions were played aloud and the written instructions were also distributed to each subject. To make subjects understand the instructions clearly, practice rounds were run before the actual experiment started. Before the actual session started, subjects practised clicking their mouses according to the experimenter’s directions to get used to how to manipulate the computers and how to understand the information shown on the screen for their decision making. They were not allowed to make free decisions until the actual round started. 5.2. Design of experiments and prediction of their results The parameters t, w, b, c employed in the experiments are shown in Table 2. This table also shows the critical value q* n. As we stated, at the critical probability q* the expected utilities of the two strategies, i.e., the R strategy and the S strategy, are equal. n is the number of participants (in our experiments it is 20). Note that t > q* n in five experiments, E1, E2, E3, E5, E8 and that the reverse is true in the three other experiments, E4, E6 and E7. Since the probability q* depends on t as well as on b, c and n, the critical value q* n is not the same value as the threshold number t. The relationship between q* and t is presented in previous Section 4 as Proposition 3. 5.3. Examination of experimental results The 1st round tattlers in Table 2 represent the number of participants who choose to take the R strategy at the first round.9 The expectation based on the comparison of the critical value q* n and 7
As for the advantages of using the World Wide Web, see Shibata et al. (1997). The materials used in the experiments will be sent upon request. 9 In the first round of E8, we told the participants to make their decisions as if there had been 13 players who took the R strategy before the first round. Thus, we treat the ratio of 13 over n as the initial probability q0 with which players think that the others choose the R strategy at the first time for this experiment. 8
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Table 2 Design of experiments, prediction and actual results Experiments t threshold w endowment b disutility c cost
q* n critical 1st round valuea tattlers
Predicted movementb
E1
15
120
110
10
11.7
14
E2 E3 E4 E5
18 15 15 15
135 120 150 120
125 110 30 110
10 10 120 10
14.9 11.7 18.4 11.7
13 18 9 16
→
E6 E7 E8
15 15 15
150 150 120
30 30 110
120 120 10
18.4 18.4 11.7
11 5 13
a b c
Actual movementc
The critical values where n = 20. The predicted movement of the number of tattlers based on the 1st round tattlers. The actual movement of the number of tattlers in experiments.
the number of the 1st round tattlers shows the graphical movements of tattling strategies which the adjustment process predicts. On the other hand, the last column gives the actual movements of the tattlers in the experiments. We can derive the following observations by examining the experimental results. E2, E4, E6 and E7 predict the decreasing movements of tattlers and the experiments, E3 and E5, the increasing movements of tattlers, given q* n and the number of tattling participants for the first round. For example, in E2, the number of tattlers in the first round is 13 and the critical value q* n is 14.9, thus, the decreasing movements of tattlers are expected. In E3 and E5, we expect the increasing movements, as the number of tattlers in the first round is greater than 11.7 which is q* n. In E5 but not E3, the increasing movement over the rounds is seen. Since the very high initial number, i.e., 18, is reached in E3, the clear increasing movement of tattlers by round may be difficult to be observed, since the maximum possible number for tattlers is 20.10 The important cases are E1 and E8 where the number of tattlers in the initial round lies less than the threshold number t but greater than the critical value, q* n. Our model predicts that the number of tattlers increases, since the initial number is greater than the critical value, q* n. However, common sense may suggest the decreasing movement of tattlers as the rounds continue because the number of tattlers in the initial round does not reach the threshold number t. The experimental results support our model and thus, imply the relevance of the adjustment process.11 Our model in Section 4 shows that if the value of the critical probability q* is small, the probability that the initial probability q0 is greater than q* is large, thus, the possibility of the bystanders choosing the R strategy is large at the initial round. (Note that q0 is an exogenous constraint.) The results of our experiments coincide with these theoretically expected results. Let 10 We implemented computer simulations and a statistical analysis to examine relevance further (Shibata et al., 2000). The results strengthened our conclusion. In order to investigate more detail about the round by round decisions, we superimposed situations with different types of players on computer programs. But the number of expected utilitymaximizing players dominated in each simulation. The result of the Mann–Whitney test suggests that the experimental results are derived mainly from the behaviors of the expected payoff maximizers. 11 It looks as though that essentially there are two treatments to the experiment: a high value b and low value c and a low value b and a high value c. And a quick analysis may suggest that the relative value of c is vital. However, this quick analysis cannot explain a contrary movement in E2.
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us call E4, E6 and E7 Group A and E1, E3 and E5 Group B.12 The critical probability q* for Group A is given in Table 2 as 0.92 (=18.4/20) and for Group B 0.58 (=11.7/20). For Group A, the average number of players who choose the R strategy in the first round is 8.3, while for Group B, it is 16.0. The difference is statistically significant at the 5% level. These observations reasonably proof the actual relevance of the initial round and the subsequent adjustment process. 6. Conclusions and some policy implications We model the apathy of bystanders as a non-cooperative game and show that there are exactly two symmetric pure strategy equilibria, all tattle and none tattle, and exactly one symmetric mixed strategy equilibrium. Under monotonic dynamic process, the pure strategy equilibria are both stable and the mixed strategy equilibrium is unstable. We also verify the theoretical adjustment process by experiments. The following policy implications are obtained. This analysis implies that in classroom and other communities such as work areas, prisons, the military and underclass ghettos, grouping the members into smaller groups may help to induce concerned and helpful behavior among members of these communities. That is, from Proposition 3-3, we see that the smaller the size of a community, the larger the probability of resolving the continuing illegal activities. And this result is not based on the general belief that a supervisor or a teacher can supervise the smaller group or classroom more easily than the larger group or classroom. Rather, the players choose to act as concerned and helpful with their decisions. Our research also implies that the following measures encourage bystanders to choose the concerned and helpful behavior. (i) The authorities, firms and schools are encouraged to advocate to members of communities that hidden crime, extortion and bullying are illegal activities and violations of human rights.13 This may increase the value of b (negative public goods) and according to Proposition 3-1, an increase in b encourages the concerned and helpful behavior of bystanders. (ii) Another measure is that the private cost c of tattling could be made as small as possible, even negative. (iii) According to our Proposition 3-2, if the threshold number t, the minimum necessary number of tattlers, decreases, the possibility of apathetic behavior also increases. (iv) It is implied that these measures will be more effective if they are introduced as early as possible. Because it is important that the initial probability of tattling by the bystanders, q0 , exceed the critical probability q* which is the dividing probability between converting all tattle and none tattle. Acknowledgements We would like to express our special appreciation to Professor Daniel Freidman of University of California, Santa Cruz, Professor Shyam Sunder of Yale University, Professor FranciscoJavier Brana of Universidad Complutense de Madrid and Professor Tatsuyoshi Saijyo of Osaka University for their helpful comments and encouragement. We would like to acknowledge our appreciation for financial support we received from Japanese Ministry of Education and Science, Tezukayama University and Kwanseigakuin University.
12 The six experiments were selected as critical values are the same within each group but different between two groups. E8 was not included due to a reason stated in note 8. 13 International research by Ministry of Education (2000) indicates that such education at home is not available in Japan compared with five other industrialized countries.
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Appendix A. Proofs First, we define Z(q:x, y) x y qk (1 − q)y−k Z(q : x, y) ≡ k k=0 Eq. (7) can be modified as follows:
(A1)
n−1 qk (1 − q)n−1−k k=0 k n−1 qt−1 (1 − q)n−t t−1
t−2 b Z(q : t − 2, n − 1) = = Z(q : t − 1, n − 1) − Z(q : t − 2, n − 1) c
=
t−2 k=0
1 n−1 t−1
n−1 k
qk+1−t (1 − q)t−1−k ≡ g(q)
Differentiating g(q) with respect to q, we have t−2 n−1 1
g (q) = qk−t (1 − q)t−2−k (k + 1 − t) k n − 1 k=0
(A2)
(A3)
t−1 Because (k + 1 − t) in RHS of (A3) is negative, g (q) should be negative. We can also obtain lim g(q) = +∞
(A4)
lim g(q) = 0
(A5)
q→0 q→1
From g (q) < 0, (A4) and (A5), we can prove that the probability q* satisfying (A2) uniquely exists for given parameters {b, c, n, t} from the intermediate value theorem. Appendix B. Proofs B.1. Proof of Proposition 3-1 The mixed strategy Nash equilibrium q* satisfies b/c = g(q* ). From the relation, we have = 1/q (q∗ )c < 0, dq∗ /dc = −g(q∗ )/q (q∗ )c > 0.
dq∗ /db
B.2. Proof of Proposition 3-2 Let us define Pd (q) by Pd (q : t, n − 1) ≡
Z(q : t − 1, n − 1) − 1. Z(q : t − 2, n − 1)
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Eqs. (B1) and {(B2)} determines the Nash equilibrium when there are n players and the threshold is t{t + 1}. c = Pd (q : t, n − 1) b c = Pd (q : t + 1, n − 1) b
(B1) (B2)
Because Pd (q:t, n − 1) = 1/g(q) and g (q) < 0, Pd (q) is increasing in q. Denote the solution of (B1){(B2)} by q* (t), {q* (t + 1)}. If Pd (q:t, n − l) > Pd (q:t + l, n − l) holds for every q, we have q* (t) < q* (t + 1). The above inequality is reduced to the following inequality: Z(q : t − 2, n − 1)Z(q : t, n − 1) − [Z(q : t − 1, n − 1)]2 < 0
(B3)
By writing p = 1 − q, (B3) can be modified as (n − 1)! n−1−t t−1 q [(n − t)q − tp]Z(q : t − 1, n − 1) p t!(n − t)! n−1 n−1 − p2n−2t−1 q2t−1 < 0 t−1 t
(B4)
The term [(n − t)q − tp]Z(q:t − 1, n − 1) in (B4) is reduced to (B5) by using the relation: t−1 t−1 n−1 n−1 n−1 n−k k n−k k t p q = t p q +t pn q 0 k k 0 k=0 k=1 t−2 n−1 t pn−1−k qk+1 − tpn . = k + 1 k=0 [(n − t)q − tp]Z(q : t − 1, n − 1) =
t−2
(n − 1)! {(n − t)(k + 1) − t(n − 1 − k)}pn−1−k qk+1 (k + 1)!(n − 1 − k)! k=0 n−1 +(n − t) pn−t qt − tpn t−1
(B5)
Here, we define Q by Q≡
t−2 k=0
(n − 1)! {(n − t)(k + 1) − t(n − 1 − k)}pn−1−k qk+1 (k + 1)!(n − 1 − k)!
The term (n − t)(k + 1) − t(n − 1 − k) is negative, since k + 1 < t. Substituting RHS of (B5) into (B4), we can transform LHS of (B4) Q
(n − 1)! n−1−t t−1 (n − 1)! q − p p2n−t−1 qt−1 t!(n − t)! (t − 1)!(n − t)!
which is negative. Therefore, (B3) holds.
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B.3. Proof of Proposition 3-3 Under the condition of t/n = v, an increase of t by 1 must be accompanied by an increase of n by 1/v. Since t and n are integers, the value of 1/v should also be an integer. Thus, we assume from here on that 1/v is integer. Eq. (B6){(B7)} determines the Nash equilibrium when the parameters are (t, n){(t + 1, n + 1/v)} and N = n − 1. c (B6) = Pd (q : t, N) b 1 c = Pd q : t + 1, N + (B7) b v Note that Pd (q) is increasing in q. Denote the solution of (B6){(B7)} by q* {t, N), {q* {t + 1, N + 1/v)}. If Pd (q:t, N) > Pd (q:t + 1, N + 1/v) holds for all q, we obtain q* (t + 1, N + 1/v) > q* (t, N). The above inequality is equivalent with 1 Z(q : t, N + 1/v) Z(q : t − 1, N) − Pd (q : t, N) = − I ≡ Pd q : t + 1, N + v Z(q : t − 1, N + 1/v) Z(q : t − 2, N) 1 1 = Z q : t, N + Z(q : t − 2, N) − Z(q : t − 1, N)Z q : t − 1, N + <0 v v (B8) Let’s define p = 1 − q. From (A1), the term I in (B8) is modified as t t−2 N + 1/v N N+1/v−k k p pN−k qk q I= k k k=0 k=0 t−1 t−1 N N + 1/v N−k k p pN+1/v−k qk − q k k k=0 k=0
(B9)
For simplicity, we introduce two terms as t−1 N + 1/v pN+1/v−k qk A≡ k k=0 t−1 N pN−k qk B≡ k k=0 The term I is further reduced to N N + 1/v (N+1/v)−t t N−(t−1) t−1 B− p − AB q q I = A+ p t−1 t
t−1 t−1 N + 1/v N N + 1/v N−t+1 t−1 N+1/v−k−1 k+1 =p p q q − t k k k=0 k=0
N + 1/v N N pN+1/v−t qt (B10) × pN+1/v−k qk − t t−1 t−1
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where M≡
t−1
k=0
N + 1/v t
N k
pN+1/v−k−1 qk+1 −
t−1
k=0
N + 1/v k
N t−1
pN+1/v−k qk
We transform M as follows: t−2 N + 1/v N N + 1/v N+1/v−k−1 k+1 M≡ p q + t k t k=0 t−1 N + 1/v N N pN+1/v−k qk pN+1/v−(t−1)−1 q(t−1)+1 − k t − 1 t−1 k=1 t−2 N + 1/v N N + 1/v N N+1/v 0 pN+1/v−k−1 qk+1 q = − p t k 0 t−1 k=0 t−2 N N + 1v pN+1/v−(k+1) qk+1 − t − 1 k + 1 k=0 N N + 1/v N pN+1/v + pN+1/v−t qt − (B11) t−1 t t−1 We rearrange the first and second terms in (B11) and introduce Q Q≡
t−2
p
N+1/v−k−1 k+1
k=0
q
N + 1/v t
N k
−
N + 1/v k+1
N t−1
(B12)
By using Q, we can rewrite the term I.
N + 1/v N N N−t+1 t−1 N+1/v−t t I=p Q+ p q q − pN+1/v t t−1 t−1
N N N + 1/v N+1/v−t t N+1/v N−t+1 t−1 Q− p − p q =p q t−1 t−1 t (B13) Because Q < 0 from Lemma 1, the term I should be negative, and (B8) also holds. Finally, we prove the following lemma. Lemma 1. Proof Q < 0. To prove that Q < 0, it is sufficient to establish the following inequality. N + 1/v N N + 1/v N − <0 t k k+1 t−1
(B14)
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where 1 and t are positive integers, v 0≤k ≤t−2
N, k,
(B15) (B16)
We can modify (B14) as follows: N + 1/v N k+1 t−1 > 1 J≡ N + 1/v N t k J is reduced to J= =
t(N − t + 2)(N − t + 3)· · ·(N − k) (k + 1)(N + 1/v − t + 1)(N + 1/v − t + 2)· · ·(N + 1/v − (k + 1)) t(t − 1)(t − 2)· · ·(k + 3)(k + 2) · (t − 1)(t − 2)· · ·(k + 2)(k + 1) (N − t + 2)(N − t + 3)· · ·(N − k) (N + 1/v − t + 1)(N + 1/v − t + 2)· · ·(N + 1/v − (k + 1))
=
t−(k+2) i=0
(t − i)(N − t + 2 + i) (t − 1 − i)(N + 1/v − t + 1 + i)
(B17)
From (B15) and (B16), both the denominator and numerator of (B17) are positive. It is easily shown that (t − i)(N − t + 2 + i) > (t − 1 − i)(N + 1/v − t + 1 + i) for all i and the numerator is greater than the denominator by 1/v + i/v. Thus, we have proven that J exceeds 1. References De Jasay, A., 1989. Social Contract, Free Ride: A Study of the Public Goods Problem. Oxford University Press. Friedman, D., Fung, K.C., 1996. International trade and internal organization of firms: an evolutionary approach. Journal of International Economics 41, 113–137. Gardner, R., Ostrom, E., Walker, J., 1990. The nature of common-pool resource problems. Rationality and Society 2, 141–147. Hirshleifer, J., 1983. From weakest-link to best shot: the voluntary provision of public goods. Public Choice 41 (3), 371–386. Hoel, H., Sparks, K., Cooper, C.L., 2001. The cost of violence/stress at work and the benefits of a violence/stress-free working environment. Report Commissioned by the Internal Labour Office, Geneva. Martino, V.D., 2002. Violence at workplace: the global response. Asian-Pacific Newsletter on Occupational Health and Safety 9 (1), 4–7. Ministry of Education, 2000. Kodomono doutokukan ya kateideno shituke ni kansuru kakoku hikakuchousa. (International Research of Education at Home Among Five Countries, Germany, Japan, Korea, UK and US). Tokyo. Ministry of Education, 1994 and 1996. Seito shidoujyo no shyomondai no genjou to monbushou no seisaku ni tuite. (Some issues in guiding students and Ministry of Education’s policy in Japan) Tokyo. Olweus, D., Smith, P.K., et al., 1999. In: Morita, Y. (Ed.), Nature of School Bullying: A Cross-National Perspective. Routledge, London. Ort, C.V., 1999. Bearing Down on the Bully. The Rotarian 174 (3), 22–24. Palfrey, T.R., Rosenthal, H., 1984. Participation and provision of discrete public goods: a strategic analysis. Journal of Public Economics 24, 171–193.
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Rigby, K., Slee, P.T., 1991. Bullying among Australian school children: reported behavior and attitudes toward victims. Journal of Social Psychology 131 (5), 615–627. Runge, C.F., 1984. Institutions and the free rider: the assurance problem in collective action. Journal of Politics 46, 151–181. Sandler, T., 1992. Collective Action: Theory and Applications. University of Michigan Press. Schelling, T.C., 1973. Hackey helmets, concealed weapons, and daylight saving: a study of binary choices with externalities. Journal of Conflict Resolution 17, 381–428. Shibata, A., Mori, T., Okamura, M., Soyama, N., 1997. WWW wo tsukatta Game System (The Game System with WWW). Working Paper Series, School of Policy Studies, Kwansei Gakuin University, No. 5, pp. 1–37. Shibata, A., Mori, T., Okamura, M., Soyama, N., 2000. An Economic Analysis of Non-Good Samaritan Behavior: Theory and Experiment. Working Paper Series, School of Policy Studies, Kwansei Gakuin University, No. 17, pp. 1–31. Weibull, J.W., 1995. Evolutionary Game Theory. The MIT press.
An economic analysis of apathetic behavior: Theory and experiment夽 Aiko Shibata a,∗ , Toru Mori b , Makoto Okamura c , Noriko Soyama d a
d
Japanese Fair Trade Commission, Kasumigaseki, Tokyo, Japan b Faculty of Economics, Nagoya City University, Japan c Economics Department, Hiroshima University, Japan Center of Research and Development of Liberal Arts Education, Tenri University, Japan Accepted 1 December 2006
Abstract The apathy of bystanders often prevails when instances of bullying, hidden crime and extortion occur in communities such as schools, business work areas, underclass ghettos, prisons and the military. The present study models apathetic behavior as a non-cooperative game and attempts to verify this theory through experiments. Furthermore, our research suggests that the apathy of bystanders generally decrease as the number of citizens in a community decrease. In our experimental cases, if the number of members in a group decreases from 40 members to 20 members, the concerned and helpful behavior of bystanders increases by 21%. © 2007 Elsevier Inc. All rights reserved. JEL classification: C92; H41 Keywords: School bullying; Workplace bullying; Bystanders’ behavior; Tattling behavior; Non-cooperative games; Dynamic adjustment process; Public goods; Experimental economics
1. Introduction Bullying can contribute to an environment of fear and intimidation in school and other places, and has become serious matters in the world (Olweus et al., 1999). For example, according to National Conference of State Legislatures in the U.S.A., 17 states enacted school safety laws
夽 ∗
Authors’ names are listed by seniority. Corresponding author. E-mail address: [email protected] (A. Shibata).
1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2006.12.026
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concerning bullying and student harassment between 1994 and 2004. The OECD jointly with Norwegian Government and University held an International Policy Conference on School Bullying and Violence in 2004. According to a 1996-survey of 2900 middle and high school students by the Seoul Family Court in South Korea, about 90% of the students was physically threatened by other students (Ort, 1999). However, bystanders may not be willing to tattle the illegal activities to the school authorities. Bullying is also observed at workplace. The cost of violence and bullying at work is very large as well. For example, the UK nation-wide survey of workplace bullying reported that the bullied group was found to have higher absenteeism due to bullying and, the currently bullied group was found to have on average 7 days more off work in a year than those who were neither bullied nor had witnessed bullying taking place. Those who are bullied also decide to leave their organization as a result of their experience. This example of bullying in the UK could be appropriate for other countries as well (Martino, 2002; Hoel et al., 2001). There are similar situations as bullying. In crime-ridden areas, people are extremely reluctant to testify against those whom they have observed robbing and killing because they expect that the criminals will not go to jail and will punish any witnesses that testify. Another situation may be political revolution. If you revolt and not enough others join you, you will be punished. If the number of people who revolt reaches a threshold, the revolt will succeed and revolutionaries will be rewarded rather than punished. The purpose of this paper is to address this question of why the apathy of bystanders predominates despites their compassionate feelings1 . To this end, we formulate the phenomenon of bullying in a school as a non-cooperative n person game. This game is a kind of binary choice public goods game with threshold. We will also examine the relevance of our model by experiment. The rest of paper is organized as follows: Section 2 formulates the model. Section 3 derives three type Nash equilibria and examines the stability of the equilibrium by introducing a dynamic adjustment process. We consider policy implications in Section 4. Experiment study is reported in Section 5. Final section gives concluding remarks. 2. The Model We consider bullying in school. A schoolyard has a bully who does bad stuff that is observed by the n other children but not by the teacher. The teacher will punish the bully if at least t of the children tattle on him. If she does not tattle and fewer than t other people tattle, she pays a cost b of being annoyed by continued harassment from the bully in a schoolyard. If she does tattle and fewer than t − 1 other people tattle, she should pay not only the cost b but an extra cost c (perhaps the bully gives her extra harassment as punishment for her tattling). If at least t people in the class tattle, then the bully is stopped. In this case nobody in the class has to pay the cost b, nor do those who tattled have to pay any cost for tattling (perhaps the bully is sent to reform school and cannot punish the squealers). The unusual, but perhaps often realistic, feature
1 Bystanders may not be willing to report the illegal activities to the school authorities. The reason for this may be explained as follows. The students’ recognition of the illegal instances which occur in their classroom is higher by several percentage points than the teachers’ recognition of the same instance (Rigby and Slee, 1991). But this number reverses itself when the actual reports of instances are counted. According to a survey of 20,000 students carried out by the Ministry of Education in Japan (1994/1996), the rate of discovery of instances of bullying by teachers is two to three times higher than the rate of discovery based on reports of students.
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Table 1 A representative player’s payoffs when she selects two strategies, the not tattling strategy S and the tattling strategy R Numbers of other tattlers
0
S (not-tattle) R (tattle)
w−b w−b−c
···
t−2
t−1
t
w−b w−b−c
w−b w
w w
···
n−1 w w
The top row shows the number of other people who tattle and w is the initial utility level when there is no disorderly activity in a community.
of the payoff structure is that tattling is costly only if there are not enough tattlers to guarantee enforcement.2 We model the above story as a n player non-cooperative game. The players are n students. Each player (student) has two strategies: tattling (R) and not tattling (S). There is a threshold value t (0 ≤ t ≤ n) in this game. The payoff of a player is illustrated in Table 1. We can interpret this game as follows: in this game, a status quo is a situation with continuous illegal activity (disorder) generating negative externalities. To eliminate the negative externalities by tattling can be interpreted to provide a public good. The tattling cost c corresponds to a provision cost of the public good. The strategy R(S) shows the concerned and helpful (apathetic) behavior. Then, this game belongs to n person, binary-choice, public good supply game. Sandler (1992) classifies this public good game into six categories, Prisoner’s Dilemma, Chicken, Coordination, Assurance 1 and Assurance 2. The game we define is in the category of Assurance 2, which describes the provision of a discrete public good without refunding costs of players. Minimum efforts must be provided to receive any benefit resulting from the public good and the associated costs are solely assigned to each provider (selecting the R strategy). On the other hand, costs are shared among all players in Assurance 1. The literature includes Hirshleifer (1983), De Jasay (1989), Gardner et al. (1990), Palfrey and Rosenthal (1984), Runge (1984) and Schelling (1973). The game we construct, however, differs from the Assurance game in two points. First, in our game, each player does not need to incur the costs once the negative externalities disappears in the class, while, in the Assurance 2, the cost has to be paid by a player at any situation. This makes a mirror image of the game with refund that each player is not required to pay the cost unless the public good is provided. Second, the benefit each player receives does not vary across the number of contributors, whilst this benefit rises with that of contributors in the Assurance 2. The structure of equilibrium is, however, essentially same as that of Assurance 2. 3. The Nash equilibrium and its dynamic stability First, we derive the Nash equilibrium of the game. In our game, every player is assumed to have an identical utility (payoff) function. The utility of a player depends on not only her strategy but also on the strategies chosen by the other players. The crucial determinant of a player’s utility is the number of other players who tattle the disorderly activity. The combination of a representative player’s and others’ strategies are listed in the pre-stated Table 1. Let us consider a typical player and assume that she thinks that the probability of the other players’ taking the R strategy is q. Then, 2 Here, we assume that an extra cost c of tattling is the costs of possible retaliation. Because time cost for tattling is very small as tattling could be performed at any time when a tattler chooses at his convenience and a traveling cost to a place for tattling is also very small as tattling happens at a place where one visits regularly, such as work places and school buildings.
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the expected utility of this player when she chooses the strategy R(S) is given by ER (q)[ES (q)]. ER (q) = (w − b − c)Z(t − 2) + w [1 − Z(t − 2)] = w − bZ(t − 2) − cZ(t − 2)
(1)
ES (q) = (w − b)Z(t − 1) + w [1 − Z(t − 1)] = w − bZ(t − 1)
(2)
where Z(k) is the sum of probabilities that the number of other players who take the R strategy is less than or equal to k. For example, t−2 n−2 Z(t − 2) = qk (1 − q)n−2−k (3) k k=0 Z(t − 1) =
t−1 k=0
n−1 k
qk (1 − q)n−1−k
By substituting these values into (1) and (2), we obtain as follows: t−2 n−2 qk (1 − q)n−2−k ER (q) = w − (b + c) k k=0 Es(q) = w − b
t−1 k=0
n−1 k
(4)
(5)
qk (1 − q)n−1−k
(6)
Since Z(t − 1) > Z(t − 2) from (3) and (4), this difference in the two probabilities reflects the difference in the threshold number of tattlers from the member’s point of view. If she tattles to the authority, the threshold number t − 1, is the minimum required number of tattling by others necessary to resolve the continuous illegal activity in the community. If she does not tattle to the authority, the threshold number remains t. In Eq. (5), the level of utility w is decreased both by the expected amount b of the negative externality and by the expected private cost c. In Eq. (6), the level of utility w is decreased only by the expected amount b of the negative externality. However, the corresponding probabilities in the two equations are different. At the mixed strategy equilibrium, both expected utilities should be equal, i.e., ER (q(t)) = ES (q(t)). By rewriting this relationship, we obtain the following: b Z(t − 2) = . c Z(t − 1) − Z(t − 2)
(7)
Then, we have the following proposition. Proposition 1. There are exactly two pure strategy Nash equilibria: all players tattle E(1) and none tattle E(0). And one symmetric mixed strategy equilibrium E(q* ) exists. Proof. Appendix A is referred.
Let us write the expected utility at equilibria E(0), E(1) and E(q* ) as U(0), U(1) and U(q* ), respectively. We then have the following relation between them: U(0) < U(q∗ ) < U(1)
(8)
Eq. (8) shows that the equilibrium E(1) where all players tattle dominates the other equilibria in the Pareto sense.
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Fig. 1. The structure of the players’ expected payoffs.
The proof of this proposition may become more transparent by showing Fig. 1 and then justifying its qualitative character. Fig. 1 shows the graphic expression of ER (q) and ES (q) which represent the player’s expected utilities from tattling and not tattling, respectively, and the graphic expression of them was obtained by applying specific numbers to variables in Eqs. (5) and (6) for ER (q) [ES (q)].3 The vertical axis measures the expected utilities and q is a probability of the other players’ tattling. For example, if the probability of tattling by the other players is greater than q* , then, ER (q) > ES (q), thus, the player is better off to tattle. On the other hand if the probability is less than q* , the reverse is true. ER (q) < ES (q) and the player is better off not to tattle. E(1) and E(0) are two pure strategy equilibria, all tattle and none tattle, respectively. The dynamic process converges to E(1)[E(0)] when the initial value of q, q0 , is larger (less) than q* . Thus, we have two stable equilibria, E(0) and E(1), and one unstable mixed strategy equilibrium, E(q* ). Let us call q* the critical probability as it is the dividing probability between converting all tattlers and none tattlers. By noting the following four facts, the qualitative character of Fig. 1 is justified: 1. 2. 3. 4.
ER (0) < ES (0). ER (1) = ES (1). There is exactly one q* ∈ (0,1) such that ER (q* ) = ES (q* ). The functions ER and ES are continuous.
From this, it follows trivially that ER (q) < ES (q) for all q < q* and ER (q) > ES (q) for all q ∈ [q* ,1), The only step of the above proof that is not immediately obvious is 3, and we have a proof 3 Fig. 1 is drawn from Eqs. (5) and (6) by assuming w = 150, b = 110, c = 10, n = 20, t = 15. These values are the values used for experiments.
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in Appendix A which could be read after the following section explaining our game in more detail. Next, we investigate which equilibrium players finally choose among the three Nash equilibria in the game. To tackle this problem, we formulate a dynamic adjustment process of the players’ strategies. Generally, the dynamic adjustment process indicating the change of q(m) can be formulated as Eq. (9) where t shows time and Φ(x) is a sign-preserving function, that is, Φ(x) 0 if and only if x 0. dq(m) = Φ ER (q(m)) − ES (q(m)) dt
(9)
We explain how the dynamic process (9) works. Suppose that two types of players, those who choose the pure strategy R, S, respectively, coexist in this game. We denote the number of players choosing the R strategy at a particular time t by r(m), 0 ≤ r(m) ≤ n. Each player can calculate her expected utility if she chooses pure strategy, R [S], for given r(m) as denoted by ER (r(m)) [ES (r(m))]. Since each player is assumed not to be perfectly-rational, she takes a time to change her strategy from S to R even though ER (r(m)) exceeds ES (r(m)) and vice versa. Consequently, only a fraction of players with less favorable strategies can switch their strategies to favorable one. Based on this idea, we can construct full dynamic adjustment process for three possible cases. (Case 1) :
ER (r(m)) > ES (r(m))
Players who choose the R strategy stick to this strategy because they enjoy the highest utility from the R strategy. A fraction of [n − r(m)] players who select the S strategy switch to the R strategy which gives them higher utility. Then the dynamic adjustment process in this case is given by dr(m) = β [n − r(m)] dt
(10)
where β is the fraction of players with changing strategy, 0 ≤ β ≤ 1. Since we can assume that q(m) = r(m)/n and n is fixed, the above process is rewritten as dq(m) = β 1 − q(m) dt
(11)
which belongs to (10). (Case 2) :
ER (r(m)) < ES (r(m))
Players with the S strategy continue to adopt their strategies. Since players with the R strategy are not satisfied, β fraction of them changes to the more advantageous strategy S. The adjustment process is defined by dr(m) = −βr(m), dt
(12)
that can be transformed to as follows: dq(m) = −βq(m) dt (Case 3) :
ER (r(m)) = ES (r(m))
(13)
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All players do not change their chosen strategies because they are indifferent between two strategies. Then, we have dr(m)/dm = dq(m)/dm = 0. Combining the adjustment processes (11) and (13) which belong to (9), we can construct full dynamic adjustment processes. Then, we establish the following: Proposition 2. Under the above dynamic adjustment process, (9), both the pure strategy Nash equilibria are globally stable, while the mixed strategy is globally unstable. Friedman and Fung (1996), for example, formulate the process which determines the distribution of two types of firms (American and Japanese) when each firm engages in Cournot competition in an autarkic or international market. Weibull (1995) provides an excellent survey and exposition of evolutionary game theory. 4. Policy implications In this section, we shall first discuss some theoretical issues and then present some policy implications for increasing bystanders’ concerned and helpful behavior. In our game-theoretical model presented in Section 3, we can show that the initial state of the game determines the final equilibrium. Since q0 is the probability at which players predict that the others choose the R strategy initially and is considered to be uniformly distributed, it is natural to consider q0 as an exogenous constraint. Thus, we assume that we cannot control the distribution of q0 . Then, the remaining procedure involves realizing the optimal equilibrium E(1) by decreasing the critical probability q* to as low a level as possible. The smaller the value of q* , the larger the probability that q0 is greater than q* , thus, the larger the possibility of the bystanders choosing the R strategy is at the initial round. This theoretical expectation is shown later to coincide with our experimental results. We examine what measures we need to lower q* . Since q* depends on four parameters {t, c, b, n}, it can be written as q* = q* (t, c, b, n). Thus, it is necessary to change the values of these four parameters in order to decrease q* . The relationship between q* and the other parameters can be specified by the following propositions. Proposition 3. 3-1 An increase in b or a decrease in c, hence an increase in b/c, reduces the value of q* . 3-2 A decrease in t decreases the value of q* . 3-3 Reducing n while keeping t/n unchanged, decreases the value of q* . Proof. See Appendix B.
Proposition 3-2 shows the effects of t on q* when n is given, while Proposition 3-3 illustrates the effects of n on q* when both n and t change, keeping t/n constant. Proposition 3-3 is based on the idea that it will be natural to think that the threshold number t changes proportionally to the number of members in a community, n. Then, Proposition 3-3 demonstrates that as the number of citizens in a community decreases, q* decreases, resulting in an increase in bystanders’ tattling activities. If the number of members in a group decreases from 40 members to 30 members and to 20 members, the critical probability, q* , decreases and the probability of resolving the continuing illegal activities in a community rises. For example, in our experimental cases, if the number of members in a group decreases from 40 members to 20 members, the concerned and helpful
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Fig. 2. Expected payoffs in the case where c is a negative value.
behavior of bystanders increases by 21% on average in our experimental cases.4 This decrease in the continuing illegal activities is not caused simply by the fact that it is easier for a supervisor or a teacher to supervise a smaller group. The decrease is rather caused by the expected utility maximizing behavior of the members in the community. Proposition 3-1 indicates that measures to increase b and to decrease c rises the chances of members tattling. Proposition 3-2 also means that by reducing the threshold number of tattling t, the concerned and helpful behavior increases. In the extreme cases in which c is either negative or a very large positive value, a Nash equilibrium with mixed strategy disappears. Suppose that c is negative. This situation is illustrated by the case in which a bystander is rewarded with sufficient compensation that exceeds her damages from retaliation when she chooses to tattle. For example, let us consider the situation in which the member, having tattled the illegal instances in the group, can receive a good evaluation for conduct because of her courage and honesty and in which this reward exceeds the costs of possible retaliation. In this case, the payoff obtained from tattling, w − b − c, is larger than w − b which is the payoff obtained by not tattling. That is, as described in Fig. 2 which is drawn from Eqs. (5) and (6) and variables remain the same as Fig. 1 except c which is changed to a negative value.5 The curve representing ER (q), the expected payoff in the case of tattling, runs above the curve representing ES (q), the expected 4 When the number of members in a community decreases from 40 to 20, an increase in cooperative behavior is calculated for each experimental case. For example, for Experiment E1: given t = 15, w = 120, b = 110, and c = 10, we obtain for n = 40, q* = 0.658 and for n = 20, q* = 0.585, thus, q* is reduced by 0.214 = [(1 − 0.585)/(1 − 0.658)] − 1. By calculating similarly the decreases of q* in eight experiments are 0.214 for Experiments E1, E3, E5, E8 and 0.195 for E4, E6, E7 and 0.241 for E2. The average and standard deviation are: average = 0.210, S.D. = 1.558E−02. 5 Fig. 2 is drawn from Eqs. (5) and (6) by assuming w = 150, b = 110, c = −10, n = 20, t = 15. Note that c is a negative value and represents a reward to a tattler.
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Fig. 3. Expected payoffs in the case where c is a large positive value.
payoff in the case of not tattling, for any q ∈ [0,1), and the curves coincide with each other only when q = 1. Thus, in the case that c < 0, our adjustment process leads to the only stable Nash equilibrium E(1), where all the players act as concerned and helpful. We consider, on the contrary, the case in which c takes a very large positive value. This implies the very high cost of retaliation as described in Fig. 3 which was drawn from Eqs. (5) and (6) and by changing c to a large positive value and other variables remain the same as Figs. 1 and 2.6 The curve representing ER (q), the expected payoff in the case of tattling, is always located below the curve representing ES (q), the expected payoff in the case of not tattling, for any q ∈ [0,1) and the curves cross each other only when q = 1. Thus, in this case, except for the unusual case in which all the bystanders decide to tattle the illegal incidences at the initial point in time, our adjustment process leads us, through a continuous reduction in the probability of tattling, to the only stable Nash equilibrium E(0), where all bystanders act apathetically. 5. Experimental analysis In this section, we examine the actual relevance of the adjustment process through experiment and show that the experimental results reasonably well explain the theoretical adjustment process.
6 Fig. 3 is drawn from Eqs. (5) and (6) by assuming w = 150, b = 110, c = 1000, n = 20, t = 15. Note that c represents a large cost to a tattler.
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5.1. Design of experiments We undertook 8 experiments (E1∼E8), each of which included 20 participants per group. Each participant played two consecutive sessions with ten rounds each using score tables in the form of Table 1 in Section 2. Participants in four experiments, E1, E2, E7 and E8, were from Kwansei Gakuin University and from Tezukayama University in the equal number, while in the other experiments, there were 18 Kwansei Gakuin, 16 Tezukayama and 6 Saitama University students. All participants were inexperienced undergraduate students. In each experiment, participants in the different universities were linked via a computer network. We used the system constructed by a client-server model on a TCP/IP network in E1 and E2, while we utilized the World Wide Web in the other cases.7 We did not inform the participants that the experiments were concerned with bullying problems nor any other issues. At the beginning of each session, subjects were told that they would be paid and at the end of each round, individual decisions and earnings were revealed to each player. However, the identities of subjects and where they sat were kept confidential to guarantee anonymity among subjects. The participants were given monetary rewards equal to their total scores after 20 rounds.8 Our experimental subjects were recruited from various majors at the three universities mentioned above. Subjects were randomly assigned to a booth with partitions in front and on both sides of the desk in the laboratory. It was impossible for them to make direct contact, i.e., by talking, making eye contact, with other subjects during the session. The taped instructions were played aloud and the written instructions were also distributed to each subject. To make subjects understand the instructions clearly, practice rounds were run before the actual experiment started. Before the actual session started, subjects practised clicking their mouses according to the experimenter’s directions to get used to how to manipulate the computers and how to understand the information shown on the screen for their decision making. They were not allowed to make free decisions until the actual round started. 5.2. Design of experiments and prediction of their results The parameters t, w, b, c employed in the experiments are shown in Table 2. This table also shows the critical value q* n. As we stated, at the critical probability q* the expected utilities of the two strategies, i.e., the R strategy and the S strategy, are equal. n is the number of participants (in our experiments it is 20). Note that t > q* n in five experiments, E1, E2, E3, E5, E8 and that the reverse is true in the three other experiments, E4, E6 and E7. Since the probability q* depends on t as well as on b, c and n, the critical value q* n is not the same value as the threshold number t. The relationship between q* and t is presented in previous Section 4 as Proposition 3. 5.3. Examination of experimental results The 1st round tattlers in Table 2 represent the number of participants who choose to take the R strategy at the first round.9 The expectation based on the comparison of the critical value q* n and 7
As for the advantages of using the World Wide Web, see Shibata et al. (1997). The materials used in the experiments will be sent upon request. 9 In the first round of E8, we told the participants to make their decisions as if there had been 13 players who took the R strategy before the first round. Thus, we treat the ratio of 13 over n as the initial probability q0 with which players think that the others choose the R strategy at the first time for this experiment. 8
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Table 2 Design of experiments, prediction and actual results Experiments t threshold w endowment b disutility c cost
q* n critical 1st round valuea tattlers
Predicted movementb
E1
15
120
110
10
11.7
14
E2 E3 E4 E5
18 15 15 15
135 120 150 120
125 110 30 110
10 10 120 10
14.9 11.7 18.4 11.7
13 18 9 16
→
E6 E7 E8
15 15 15
150 150 120
30 30 110
120 120 10
18.4 18.4 11.7
11 5 13
a b c
Actual movementc
The critical values where n = 20. The predicted movement of the number of tattlers based on the 1st round tattlers. The actual movement of the number of tattlers in experiments.
the number of the 1st round tattlers shows the graphical movements of tattling strategies which the adjustment process predicts. On the other hand, the last column gives the actual movements of the tattlers in the experiments. We can derive the following observations by examining the experimental results. E2, E4, E6 and E7 predict the decreasing movements of tattlers and the experiments, E3 and E5, the increasing movements of tattlers, given q* n and the number of tattling participants for the first round. For example, in E2, the number of tattlers in the first round is 13 and the critical value q* n is 14.9, thus, the decreasing movements of tattlers are expected. In E3 and E5, we expect the increasing movements, as the number of tattlers in the first round is greater than 11.7 which is q* n. In E5 but not E3, the increasing movement over the rounds is seen. Since the very high initial number, i.e., 18, is reached in E3, the clear increasing movement of tattlers by round may be difficult to be observed, since the maximum possible number for tattlers is 20.10 The important cases are E1 and E8 where the number of tattlers in the initial round lies less than the threshold number t but greater than the critical value, q* n. Our model predicts that the number of tattlers increases, since the initial number is greater than the critical value, q* n. However, common sense may suggest the decreasing movement of tattlers as the rounds continue because the number of tattlers in the initial round does not reach the threshold number t. The experimental results support our model and thus, imply the relevance of the adjustment process.11 Our model in Section 4 shows that if the value of the critical probability q* is small, the probability that the initial probability q0 is greater than q* is large, thus, the possibility of the bystanders choosing the R strategy is large at the initial round. (Note that q0 is an exogenous constraint.) The results of our experiments coincide with these theoretically expected results. Let 10 We implemented computer simulations and a statistical analysis to examine relevance further (Shibata et al., 2000). The results strengthened our conclusion. In order to investigate more detail about the round by round decisions, we superimposed situations with different types of players on computer programs. But the number of expected utilitymaximizing players dominated in each simulation. The result of the Mann–Whitney test suggests that the experimental results are derived mainly from the behaviors of the expected payoff maximizers. 11 It looks as though that essentially there are two treatments to the experiment: a high value b and low value c and a low value b and a high value c. And a quick analysis may suggest that the relative value of c is vital. However, this quick analysis cannot explain a contrary movement in E2.
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us call E4, E6 and E7 Group A and E1, E3 and E5 Group B.12 The critical probability q* for Group A is given in Table 2 as 0.92 (=18.4/20) and for Group B 0.58 (=11.7/20). For Group A, the average number of players who choose the R strategy in the first round is 8.3, while for Group B, it is 16.0. The difference is statistically significant at the 5% level. These observations reasonably proof the actual relevance of the initial round and the subsequent adjustment process. 6. Conclusions and some policy implications We model the apathy of bystanders as a non-cooperative game and show that there are exactly two symmetric pure strategy equilibria, all tattle and none tattle, and exactly one symmetric mixed strategy equilibrium. Under monotonic dynamic process, the pure strategy equilibria are both stable and the mixed strategy equilibrium is unstable. We also verify the theoretical adjustment process by experiments. The following policy implications are obtained. This analysis implies that in classroom and other communities such as work areas, prisons, the military and underclass ghettos, grouping the members into smaller groups may help to induce concerned and helpful behavior among members of these communities. That is, from Proposition 3-3, we see that the smaller the size of a community, the larger the probability of resolving the continuing illegal activities. And this result is not based on the general belief that a supervisor or a teacher can supervise the smaller group or classroom more easily than the larger group or classroom. Rather, the players choose to act as concerned and helpful with their decisions. Our research also implies that the following measures encourage bystanders to choose the concerned and helpful behavior. (i) The authorities, firms and schools are encouraged to advocate to members of communities that hidden crime, extortion and bullying are illegal activities and violations of human rights.13 This may increase the value of b (negative public goods) and according to Proposition 3-1, an increase in b encourages the concerned and helpful behavior of bystanders. (ii) Another measure is that the private cost c of tattling could be made as small as possible, even negative. (iii) According to our Proposition 3-2, if the threshold number t, the minimum necessary number of tattlers, decreases, the possibility of apathetic behavior also increases. (iv) It is implied that these measures will be more effective if they are introduced as early as possible. Because it is important that the initial probability of tattling by the bystanders, q0 , exceed the critical probability q* which is the dividing probability between converting all tattle and none tattle. Acknowledgements We would like to express our special appreciation to Professor Daniel Freidman of University of California, Santa Cruz, Professor Shyam Sunder of Yale University, Professor FranciscoJavier Brana of Universidad Complutense de Madrid and Professor Tatsuyoshi Saijyo of Osaka University for their helpful comments and encouragement. We would like to acknowledge our appreciation for financial support we received from Japanese Ministry of Education and Science, Tezukayama University and Kwanseigakuin University.
12 The six experiments were selected as critical values are the same within each group but different between two groups. E8 was not included due to a reason stated in note 8. 13 International research by Ministry of Education (2000) indicates that such education at home is not available in Japan compared with five other industrialized countries.
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Appendix A. Proofs First, we define Z(q:x, y) x y qk (1 − q)y−k Z(q : x, y) ≡ k k=0 Eq. (7) can be modified as follows:
(A1)
n−1 qk (1 − q)n−1−k k=0 k n−1 qt−1 (1 − q)n−t t−1
t−2 b Z(q : t − 2, n − 1) = = Z(q : t − 1, n − 1) − Z(q : t − 2, n − 1) c
=
t−2 k=0
1 n−1 t−1
n−1 k
qk+1−t (1 − q)t−1−k ≡ g(q)
Differentiating g(q) with respect to q, we have t−2 n−1 1
g (q) = qk−t (1 − q)t−2−k (k + 1 − t) k n − 1 k=0
(A2)
(A3)
t−1 Because (k + 1 − t) in RHS of (A3) is negative, g (q) should be negative. We can also obtain lim g(q) = +∞
(A4)
lim g(q) = 0
(A5)
q→0 q→1
From g (q) < 0, (A4) and (A5), we can prove that the probability q* satisfying (A2) uniquely exists for given parameters {b, c, n, t} from the intermediate value theorem. Appendix B. Proofs B.1. Proof of Proposition 3-1 The mixed strategy Nash equilibrium q* satisfies b/c = g(q* ). From the relation, we have = 1/q (q∗ )c < 0, dq∗ /dc = −g(q∗ )/q (q∗ )c > 0.
dq∗ /db
B.2. Proof of Proposition 3-2 Let us define Pd (q) by Pd (q : t, n − 1) ≡
Z(q : t − 1, n − 1) − 1. Z(q : t − 2, n − 1)
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Eqs. (B1) and {(B2)} determines the Nash equilibrium when there are n players and the threshold is t{t + 1}. c = Pd (q : t, n − 1) b c = Pd (q : t + 1, n − 1) b
(B1) (B2)
Because Pd (q:t, n − 1) = 1/g(q) and g (q) < 0, Pd (q) is increasing in q. Denote the solution of (B1){(B2)} by q* (t), {q* (t + 1)}. If Pd (q:t, n − l) > Pd (q:t + l, n − l) holds for every q, we have q* (t) < q* (t + 1). The above inequality is reduced to the following inequality: Z(q : t − 2, n − 1)Z(q : t, n − 1) − [Z(q : t − 1, n − 1)]2 < 0
(B3)
By writing p = 1 − q, (B3) can be modified as (n − 1)! n−1−t t−1 q [(n − t)q − tp]Z(q : t − 1, n − 1) p t!(n − t)! n−1 n−1 − p2n−2t−1 q2t−1 < 0 t−1 t
(B4)
The term [(n − t)q − tp]Z(q:t − 1, n − 1) in (B4) is reduced to (B5) by using the relation: t−1 t−1 n−1 n−1 n−1 n−k k n−k k t p q = t p q +t pn q 0 k k 0 k=0 k=1 t−2 n−1 t pn−1−k qk+1 − tpn . = k + 1 k=0 [(n − t)q − tp]Z(q : t − 1, n − 1) =
t−2
(n − 1)! {(n − t)(k + 1) − t(n − 1 − k)}pn−1−k qk+1 (k + 1)!(n − 1 − k)! k=0 n−1 +(n − t) pn−t qt − tpn t−1
(B5)
Here, we define Q by Q≡
t−2 k=0
(n − 1)! {(n − t)(k + 1) − t(n − 1 − k)}pn−1−k qk+1 (k + 1)!(n − 1 − k)!
The term (n − t)(k + 1) − t(n − 1 − k) is negative, since k + 1 < t. Substituting RHS of (B5) into (B4), we can transform LHS of (B4) Q
(n − 1)! n−1−t t−1 (n − 1)! q − p p2n−t−1 qt−1 t!(n − t)! (t − 1)!(n − t)!
which is negative. Therefore, (B3) holds.
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B.3. Proof of Proposition 3-3 Under the condition of t/n = v, an increase of t by 1 must be accompanied by an increase of n by 1/v. Since t and n are integers, the value of 1/v should also be an integer. Thus, we assume from here on that 1/v is integer. Eq. (B6){(B7)} determines the Nash equilibrium when the parameters are (t, n){(t + 1, n + 1/v)} and N = n − 1. c (B6) = Pd (q : t, N) b 1 c = Pd q : t + 1, N + (B7) b v Note that Pd (q) is increasing in q. Denote the solution of (B6){(B7)} by q* {t, N), {q* {t + 1, N + 1/v)}. If Pd (q:t, N) > Pd (q:t + 1, N + 1/v) holds for all q, we obtain q* (t + 1, N + 1/v) > q* (t, N). The above inequality is equivalent with 1 Z(q : t, N + 1/v) Z(q : t − 1, N) − Pd (q : t, N) = − I ≡ Pd q : t + 1, N + v Z(q : t − 1, N + 1/v) Z(q : t − 2, N) 1 1 = Z q : t, N + Z(q : t − 2, N) − Z(q : t − 1, N)Z q : t − 1, N + <0 v v (B8) Let’s define p = 1 − q. From (A1), the term I in (B8) is modified as t t−2 N + 1/v N N+1/v−k k p pN−k qk q I= k k k=0 k=0 t−1 t−1 N N + 1/v N−k k p pN+1/v−k qk − q k k k=0 k=0
(B9)
For simplicity, we introduce two terms as t−1 N + 1/v pN+1/v−k qk A≡ k k=0 t−1 N pN−k qk B≡ k k=0 The term I is further reduced to N N + 1/v (N+1/v)−t t N−(t−1) t−1 B− p − AB q q I = A+ p t−1 t
t−1 t−1 N + 1/v N N + 1/v N−t+1 t−1 N+1/v−k−1 k+1 =p p q q − t k k k=0 k=0
N + 1/v N N pN+1/v−t qt (B10) × pN+1/v−k qk − t t−1 t−1
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where M≡
t−1
k=0
N + 1/v t
N k
pN+1/v−k−1 qk+1 −
t−1
k=0
N + 1/v k
N t−1
pN+1/v−k qk
We transform M as follows: t−2 N + 1/v N N + 1/v N+1/v−k−1 k+1 M≡ p q + t k t k=0 t−1 N + 1/v N N pN+1/v−k qk pN+1/v−(t−1)−1 q(t−1)+1 − k t − 1 t−1 k=1 t−2 N + 1/v N N + 1/v N N+1/v 0 pN+1/v−k−1 qk+1 q = − p t k 0 t−1 k=0 t−2 N N + 1v pN+1/v−(k+1) qk+1 − t − 1 k + 1 k=0 N N + 1/v N pN+1/v + pN+1/v−t qt − (B11) t−1 t t−1 We rearrange the first and second terms in (B11) and introduce Q Q≡
t−2
p
N+1/v−k−1 k+1
k=0
q
N + 1/v t
N k
−
N + 1/v k+1
N t−1
(B12)
By using Q, we can rewrite the term I.
N + 1/v N N N−t+1 t−1 N+1/v−t t I=p Q+ p q q − pN+1/v t t−1 t−1
N N N + 1/v N+1/v−t t N+1/v N−t+1 t−1 Q− p − p q =p q t−1 t−1 t (B13) Because Q < 0 from Lemma 1, the term I should be negative, and (B8) also holds. Finally, we prove the following lemma. Lemma 1. Proof Q < 0. To prove that Q < 0, it is sufficient to establish the following inequality. N + 1/v N N + 1/v N − <0 t k k+1 t−1
(B14)
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where 1 and t are positive integers, v 0≤k ≤t−2
N, k,
(B15) (B16)
We can modify (B14) as follows: N + 1/v N k+1 t−1 > 1 J≡ N + 1/v N t k J is reduced to J= =
t(N − t + 2)(N − t + 3)· · ·(N − k) (k + 1)(N + 1/v − t + 1)(N + 1/v − t + 2)· · ·(N + 1/v − (k + 1)) t(t − 1)(t − 2)· · ·(k + 3)(k + 2) · (t − 1)(t − 2)· · ·(k + 2)(k + 1) (N − t + 2)(N − t + 3)· · ·(N − k) (N + 1/v − t + 1)(N + 1/v − t + 2)· · ·(N + 1/v − (k + 1))
=
t−(k+2) i=0
(t − i)(N − t + 2 + i) (t − 1 − i)(N + 1/v − t + 1 + i)
(B17)
From (B15) and (B16), both the denominator and numerator of (B17) are positive. It is easily shown that (t − i)(N − t + 2 + i) > (t − 1 − i)(N + 1/v − t + 1 + i) for all i and the numerator is greater than the denominator by 1/v + i/v. Thus, we have proven that J exceeds 1. References De Jasay, A., 1989. Social Contract, Free Ride: A Study of the Public Goods Problem. Oxford University Press. Friedman, D., Fung, K.C., 1996. International trade and internal organization of firms: an evolutionary approach. Journal of International Economics 41, 113–137. Gardner, R., Ostrom, E., Walker, J., 1990. The nature of common-pool resource problems. Rationality and Society 2, 141–147. Hirshleifer, J., 1983. From weakest-link to best shot: the voluntary provision of public goods. Public Choice 41 (3), 371–386. Hoel, H., Sparks, K., Cooper, C.L., 2001. The cost of violence/stress at work and the benefits of a violence/stress-free working environment. Report Commissioned by the Internal Labour Office, Geneva. Martino, V.D., 2002. Violence at workplace: the global response. Asian-Pacific Newsletter on Occupational Health and Safety 9 (1), 4–7. Ministry of Education, 2000. Kodomono doutokukan ya kateideno shituke ni kansuru kakoku hikakuchousa. (International Research of Education at Home Among Five Countries, Germany, Japan, Korea, UK and US). Tokyo. Ministry of Education, 1994 and 1996. Seito shidoujyo no shyomondai no genjou to monbushou no seisaku ni tuite. (Some issues in guiding students and Ministry of Education’s policy in Japan) Tokyo. Olweus, D., Smith, P.K., et al., 1999. In: Morita, Y. (Ed.), Nature of School Bullying: A Cross-National Perspective. Routledge, London. Ort, C.V., 1999. Bearing Down on the Bully. The Rotarian 174 (3), 22–24. Palfrey, T.R., Rosenthal, H., 1984. Participation and provision of discrete public goods: a strategic analysis. Journal of Public Economics 24, 171–193.
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